On convergence of finite difference schemes for generalized solutions of elliptic differential equations

ON CONVERGENCE OF FINITE DIFFERENCE SCHEMES FOR GENERALIZED SOLUTIONS

OF ELLIPTIC DIFFERENTIAL EQUATIONS

s.

H.-J.

1. Introduction

Let n be a rectangular domain in

with boundary on. ^{We con-}

sider finite difference approximations for the generalized solutions of elliptic differential equations of the form

(1.la) A(x)u(x)

f(x), x

n,

with boundary condition

(Llb) u(x)=O,

on.

Here A(x) is a second order, self-adjoint elliptic operator with smooth coefficients which has the following form

2

0 0

A(x)u = - L

^{(a1q(x) 0: ) + a(x)u.}

~q=l ^q

Approximate solutions and error estimates for (1.1) have been ob- tained through energy arguments using Taylor's Theorem. This tra- ditional approach to the study of the rate of convergence requires a high degree of smoothness for the exact solutions. Cosequently, Tay- lor's Theorem is not the natural framework in which to establish orders of convergence in weaker norms for finite difference approximations of non-smooth solutions.

Received September 21, 1992.

The first author is supported by KOSEF under Grants 921-0100-011-1 and 91- 08-00-01. The second author is supported by KOSEF under Grant 91-08-00-01.

58

s.

In a celebrated paper on exact finite difference approximations for the generalized solutions of two point boundary value problems, Tikh- onov and Samarskii[lO], have obtained rates of convergence which are compatible with the smoothness of their solutions. But their meth- ods do not apply to the multidimensional case. Recently, there have been some results on the approximation of the generalized solutions for linear prabolic and hyperbolic partial differential equations via finite difference schemes (see Lazarov et al. [7]-[8], Jovanovic et al. [5]-[6], and Pani et al. [9]).

The key to the above approaches is to compare the exact solution with a suitable mollified approximation such as the average values on cells around grid points (instead of point estimates). Such averaging can be defined using the Steklov mollifier. The resulting comparisons of the exact and approximate solutions with the mollified approxima- tion yield sharp orders of convergence in an elegant manner. Similar procedures are utilized here.

In the present paper, we investigate rates of convergence of finite difference schemes for the approximate solution of (1.1). As for the finite element method, we obtain orders of convergence compatible with the smoothness of the solution. A discrete projection technique is then introduced to reduce the regularity to that of the generalized solution.

The preliminary material is -given in Section 2. In Section 3, a dis- crete schem:e forCl.1}is analyzed,and stability results for a modified scheme are derived, which yield the required error estimates in the discrete L

and HI norms. Nitche's technique is applied to the dis- crete scheme, and the error estimates with reduced regularity on u are derived, which yield O(h

1 <

S 2, convergence in the discrete L2- norm.

2. Preliminaries

We may assume, without loss of generality, that the domain n is the unit square in R

For the numerical solution of (1.1), we select a mesh of width h = k, ^where M is a positive integer, and cover fi = n ^u an

with a square grid of mesh points

= (ih,jh), for i,j = ^{0,1, ...} ,M.

Let n

=

n} and an

=

an}. We can cover

the whole of R2 with such a square grid, and will denote it by Ri.

For any function

defined on

we adopt the following notation:

for x

R.i and 1=1,2,

and

o ( )

v(x+he,)-v(x)

v

zv x = h '

v(x)-v(x-hez)

vZv x =

h '

where ez is the I-th unit vector in

The Steklov mollifiers

defined in the following manner:

where

s = s;si with sl = Sis" ^1=1,2,

sttjJ(x) = 1

tjJ(x+shez)ds, S,tjJ(x) = L: tjJ(x+shez)ds.

The operators st commute, and the following relations hold

sltjJ(x) = 1° ^{(1 + s)tjJ(x + shez) ds + t (1- s)tjJ(x + shez) ds,}

-1

10

and (2.1)

Let 'Dh denote the mesh functions defined on

which vanish out- side of Oh. For u, v

'Dh, we now introduce the discrete L

space, denoted

LI(Oh), with inner product and norm given

(w,v) =h

L ^w(x)v(x),

xEl!~

and

IIwllo,h ⁼ (w, w) t ,

60 S. K. Chung and H.-J. Kimn

respectively. Further, let

= HlCnh) denote the discrete analogue of the H1-Sobolev space with norm IIwlltk = IIwll~,h + L:~=1IlVlwIl2.

We also introduce a discerete H

-Sobolev space with norm

Z

2 2 ~ - Z

II ^w llz,k

IIwlh,h + L.J 1IV'

V' qwllo,k' l,q=l

and denote it by

Whenever there is no confusion, we write IIwll and IIwlli' for

= 1,2, in place of IIwllo,h and \IWl!i,h, respectively. Throughout the paper, 11· 11

and 11 ·11

will denote the norm in the

and the Sobolev space Wm,p(n), respectively. Further, I .

denotes the semi- norm on Wm,p(n). In particular, for

2, we denote Wm,p(n) by Hm(n) .

For functions v and

which vanish on an

the following identities are easy consequences of summation by parts:

(2.2) (V',V, w)

-(v, V,W), 1= 1,2.

The basic lemma, which will be used in the following sections, are given below... Along with the Bramble--Hilbert Lemma(see, Bramble and lIilbert[l} and Dupont and S~ott[4]),the following oilinear version of it will be needed for our convergence analysis. For a proof,

refer the reader to Ciarlet[2].

LEMMA

2.1. Let

be the set

all polynomials

degree:5 [r], where [r] denotes the largest integer less than r > O. If TJ is a bounded bilinear functional on WO,p(n) x w,B,q(n), with a,p E (0,00) and p, q E [1,00] such that

TJ(U,v) = 0, 17(u, V) = 0,

VU

p[a}(n), Vv

w,B,q(n),

Vu

Wa,p(n), VV

p[,B}(n),

then there exists

positive constant C such that

117( u, v) I::; ^C I u

I v

Vu

Wa,p(n), Vv

w,B,q(n).

We shall frequently use the following inequality

(2.3) a,b ER, c > O.

We write C as a generic positive constant independent of the discretiz- ing parameters h.

3. Discrete Schemes

In this section, the stability and error analysis for a discrete scheme of (1.1) will be given. Let Ah(x) be the finite difference approximations for the operators A(x), defined by

2

Ah(X)V = -~ 2: [V,(a,q(x)VqV) + V,(a,q(x)VqV)] + S(a(x»V,

l,q=l

for x E 1"h. ^In order to minimize the regularity imposed on a( x ) in the subsequent analysis, it is necessary to work with the Steklov mollification S(a( x» in the above approximations.

Our discrete approximation U on 0

to (1.1) is now defined by

(3.1) Ah(X)U(X)

Sj(x), x

Oh,

U(x) = 0, x

aOh.

We first consider a modified difference scheme of (3.1); namely,

(3.2)

2

Ah(x)U(x) = Sj(x) + I:V,F(x),

1=1

where F, defined on Oh, vanishes on aOh.

Let H5,h = {v

Hl : v = ^Oon aOh}. On the basis of the as-

sumptions imposed on A( x), the following lemma can be verified using

summation by parts.

62 S.K. Chung and H.-J. Kimn

LEMMA 3.1. For V, WE HJ ,

we have the following inequalities:

(1) the discrete Poincare inequality: 11V1I

CL:~=1I1VlVII2, (2) (Ah(t)V, V) 2 eollVlli,

where

is

positive constant.

The stability analysis will be formulated in the discrete norm.

THEOREM 3.1. Let U be a solution of (3.2). Then there exists a constant C such that

IIUII ::; ^C[IISfl/ + I/FII]·

Proof. Forming the discrete L2 inner product between (3.2) and U, we obtain

2

(Ah(X)U,U) = (Sf,U) + L(V1F,U).

1=1 Using Lemma 3.1, we obtain

IIUII~ :S C[lI SfIlI\UI\ + 1/FllllUlh]·

An application of the inequality (2.3) now completes the proof. 0 Using Theorem 3.1, we now derive the following error estimate in e(x)

u(x) - U(x).

THEOREM 3.2. Let u and U be the solution of (1.1) and (3.1), respectively. Then the error e( x) = u( x ) - U(x) satisfies the following estimate

1 <

3.

Proof. From (1.1) and (3.1), it follows that

Following Jovanovic et al. [6], the integrand G

(x) is rewritten as

2

G

(x)

L ^{VI17lq(X) + 7J(x),}

l,q=l

where

with

(1)

+

VU) (S+

(+S2 VU)

T/lq

S, S3_1 a'qVX

q - I S3_1 a1q S,

VX q ,

and

T/

(Sa)u - S(au).

To estimate T/, we first note that

T/ = (Sa)(u - Su) + (Sa)(Su) - S(au).

As in Pani et al.[9], the Bramble-Hilbert Lemma along with Lemma 2.1 yields

And we obtain the following estimate for T/lq as in Jovanovic et al. [6]

L

1IT/lq(x)1I

~ ^Ch

u(x)IIHa(o), 1 < a ~ ^3, l,q=l

where the constant C depends on max',q IlalqIILOO(Wa-1,OO). Theorem 3.1 now completes the proof. 0

We now derive the L

estimates with order of convergence com-

patible with the regularity on the generalized solution

with reduced

regularity.

64 S.K. Chung and H.-J. Kimn

THEOREM

3.3. There exists

constant C, which depends on u, such that

Proof. For the estimation of 11elh, we take the discrete inner product of (3.1) with e and obtain

Using the estimate for G

from Theorem 3.2, we obtain

Here the constant C depends on lIullH2(o),

I

and

lalwl.oo. Now from the coercivity of the operator Ah, the required estimate follows for 11 e 111.

For the discerete £2 estimate, we define

as the solution of

(3.3)

Because of the coercivity of AA,

is a unique solution of (3.3) with appropriate regularity

(3.4)

Forming the discrete inner product between (3.3) and e, we find that (3.5) (e, e) =

= ^{(AhU -}

=

For the estimation of G

we decompose G

as in Theorem 3.2, and

obtain, with an application of Lemma 2.1,

For the estimation of L7,q=1 (7]~) ^{, 'V I«P),} we consider it term by term.

For 1 = ^{1, q} = ^2,

( (3) (

+

OU 1 - + 1 + )

7]12' 'Vl «P)

SI S2 OX2 - 2(V'2 ^U +

+

I2 )V'I«P

=

('V

[si S2"u - ~(u ^{+ u+}

~(aI2 ⁺

iP) .

Here, we have used (2.1) and V',U =

e,h). The application of (2.2) yields

!( _{7]12' 'V}

_{«P) I}

SI S2"u - 2(U + 1 + U ,-), ^{+1 2 -} 'V2 2 (1 (

+

⁺¹⁾⁾⁾ 'V

«P

~clIsi ^{S2"u -} ~(u ^{+ u+}

,-2)1I11«p1I2'

Since Si S2"u - f(u + U+1,-2) vanishes for all u E Pll an application of the Bramble-Hilbert Lemma yields

The estimator for (1J~~), ^'VI «p) is obtained in a similar manner.

For 1 = 1,

1,

Since

- u vanishes for all u E PI, an application of the Bramble-

Hilbert Lemma again yields

66 S.K. Chung and H.-J. Kimn

The estimate for (77~~), V' 24» is obtained in a similar manner.

Using Lemma 2.1, we can derive the following estimates

2 2

L ^{(77}:) , V'14» '5: Gh 2} L la,qlw"',oo lI ^u IlH"'-l(o)II4>lb,

l,q=l l,q=l

2 2

L (77~), ^{V'14» '5: Gh 2 (} L la,q!w",-l.OO )/uIH"'(0)/I4>1I2'

l,q=l l,q=l

and from the Bramble-Hilbert lemma we obtain 1(77,4»1'5: I «(Sa)(u - Su) + (Sa)(Su) - S(au), 4»1

'5: GhQ [IaILOO(O) lulH'"

+ lalw"'-l.oo(o)luIH"'-l(o»)II4>/I],

1 < a '5: 2.

It

therefore follows that

Using the regularity condition (3.4), we obtain

with C depending on !alw"'-l.00(O) and max19,q9I1a,qllw"'-1.oo(0)'

This completes the proof of Theorem 3.3. 0

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of

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~erdam,1978.

3. E. G. D'jakonov, On the stability

of

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>

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Department of Mathematics Education Seoul National University

Seoul 151-742, Korea

Department of Computer Science Ajou University

Suwon 441-749, Korea